Annette Arroyo

2021-10-02

Find the derivatives of the functions $y=1+x-4\frac{\sqrt{x}}{x}$

crocolylec

Step 1
We distribute the denominator and then write using exponents
$y=\frac{1+x-4\sqrt{x}}{x}$
$y=\frac{1}{x}+\frac{x}{x}-\frac{4\sqrt{x}}{x}$
$y={x}^{-1}+1-4{x}^{-\frac{1}{2}}$
Step 2
Then we differentiate both sides using the power rule
$y={x}^{-1}+1-4{x}^{-\frac{1}{2}}$
$\frac{dy}{dx}=-1{x}^{-2}+0-4\left(-\frac{1}{2}\right){x}^{-\frac{1}{2}-1}$
$\frac{dy}{dx}=-{x}^{-2}+2{x}^{-\frac{3}{2}}$
$\frac{dy}{dx}=-\frac{1}{{x}^{2}}+\frac{2}{{x}^{\frac{3}{2}}}$
Answer: $\frac{dy}{dx}=-\frac{1}{{x}^{2}}+\frac{2}{{x}^{\frac{3}{2}}}$

Jeffrey Jordon